Asymptotic Capacity and Optimal Precoding Strategy of Multi-Level Precode & Forward in Correlated Channels

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Asymptotic Capacity and Optimal Precoding Strategy of

Multi-Level Precode & Forward in Correlated Channels

Nadia Fawaz, Keyvan Zarifi, Merouane Debbah, David Gesbert

To cite this version:

Nadia Fawaz, Keyvan Zarifi, Merouane Debbah, David Gesbert. Asymptotic Capacity and Optimal

Precoding Strategy of Multi-Level Precode & Forward in Correlated Channels. ITW2008, May 2008,

Portugal. 5 p. �hal-00353189�

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Asymptotic Capacity and Optimal Precoding

Strategy of Multi-Level Precode & Forward in

Correlated Channels

Nadia Fawaz^∗, Keyvan Zarifi^†, Merouane Debbah^‡ and David Gesbert^∗

∗Mobile Communications Department, Eurecom Institute, Sophia-Antipolis, France {nadia.fawaz,david.gesbert}@eurecom.fr

†INRS-EMT & Concordia University, Montr´eal, Canada [email protected]

‡Alcatel-Lucent Chair on Flexible Radio, SUPELEC, Gif-sur-Yvette, France [email protected]

Abstract— We analyze a multi-level MIMO relaying system where a multiple-antenna transmitter sends data to a multiple- antenna receiver through several relay levels, also equipped with multiple antennas. Assuming correlated fading in each hop, each relay receives a faded version of the signal transmitted by the previous level, performs precoding on the received signal and retransmits it to the next level. Using free probability theory and assuming that the noise power at the relay levels - but not at the receiver - is negligible, a closed-form expression of the end-to-end asymptotic instantaneous mutual information is derived as the number of antennas in all levels grow large with the same rate.

This asymptotic expression is shown to be independent from the channel realizations, to only depend on the channel statistics and to also serve as the asymptotic value of the end-to-end average mutual information. We also provide the optimal singular vectors of the precoding matrices that maximize the asymptotic mutual information : the optimal transmit directions represented by the singular vectors of the precoding matrices are aligned on the eigenvectors of the channel correlation matrices, therefore they can be determined only using the known statistics of the channel matrices and do not depend on a particular channel realization.

I. INTRODUCTION

Relay communication systems have recently attracted much attention due to their potential to substantially improve the signal reception quality when the direct communication link between the transmitter and the receiver is not reliable. Due to its major practical importance as well as its significant techni- cal challenge, deriving the capacity - or bounds on the capacity - of various relay communication schemes is growing to an entire field of research. Of particular interest, is the capacity bounds for the systems in which the transmission, reception, or the relay levels are equipped with multiple antennas. Assuming fixed channel conditions, lower and upper bounds on the capacity of multiple-input multiple output (MIMO) two-hop relay channel have been derived in [1]. Similar bounds have also been obtained in the same paper on the ergodic capacity when the communication links undergo i.i.d. Rayleigh fadings.

For a two-hop relay system and the case when the source- relay and the relay-destination channel matrices are perfectly

known, the optimal relay precoding matrix is derived in [2]. In [3] the asymptotic capacity of MIMO two-hop relay networks has been studied when the number of relay nodes grows to infinity while the number of transmit and receive antennas remain constant. The asymptotic capacity of the MIMO multihop amplify-and-forward relay channels has been derived in [4] when all channel links experience i.i.d. Rayleigh fading while the number of transmit and receive antennas as well as the number of relays at each hop go to infinity with the same rate. The scaling behavior of the capacity of MIMO two-hop relay channel has also been studied in [5] for the case when the source and destination jointly communicate to each other through the relay layer. Other related problems have been analyzed in, for instance, [6], [7], [8], [9].

In this paper we study MIMON -hop relay communication system wherein data transmission from k0 transmit antennas tokN receive antennas is made possible throughN − 1 relay levels each of which are equipped with ki, i = 1, . . . , N − 1 antennas. In this transmission chain of N + 1 levels it is assumed that the direct communication link is only viable between two adjacent levels: Each relay receives a faded version of the multi-dimensional signal transmitted from the previous level and, after precoding, retransmits it to the next level. We consider the case where all communication links undergo Rayleigh flat fading and the fading channels in each hop (in-between two adjacent levels) may be correlated while the fading channels of any two different hops are uncorrelated.

Using the tools from free probability theory and assuming that the noise power at the relay levels, but not at the receiver, is negligible, we derive a closed-form expression for the end- to-end asymptotic instantaneous mutual information between the transmitter and the receiver as the number of antennas in all levels grows large with the same rate. This asymptotic expression is shown to be independent from the channel realizations and only depends on the channel statistics. Therefore, as long as the statistical properties of the channel matrices at all hops do not change, the instantaneous mutual information asymptotically converges to the same deterministic expression

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Z

x1 x2 xN −2 xN −1

y0 y1 y2 yN −2 yN −1 yN

k0 k1 k2 k_{N −2} k_{N −1} kN

x0 H1 H2 HN −1 HN

Fig. 1. Multi-level Relaying System

for any arbitrary channel realization. As a consequence, the so- obtained expression also serves as the asymptotic value of the end-to-end average mutual information between the transmitter and the receiver. Then we obtain the optimal singular vectors of the precoding matrices that maximize the asymptotic mutual information. It is shown that, in the asymptotic regime, the optimal singular vectors of the precoding matrices are also independent from the channel realizations and can be determined only using the known statistics of the channel matrices.

The rest of the paper is organized as follows. In section II, notations and the system model are presented. In section III, the end-to-end instantaneous mutual information in the asymptotic regime is derived, whereas the optimal singular vectors of the precoding matrices are given in section IV.

Numerical results are provided in section V and lead to the concluding section VI.

II. SYSTEMMODEL

Matrices and vectors are represented by boldface uppercase.

A^T, A^∗, A^H denote the transpose, the conjugate and the transpose conjugate of matrix A. tr(A), det(A) and kAkF = q

tr(AA^H) stand for trace, determinant and Frobenius norm of A. IN is the identity matrix of size N.

Consider Fig. 1 that shows a multi-level relaying system with k0 transmit antennas, kN receive antennas and N − 1 relaying stages. The i−th relay stage is equipped with ki

antennas. We assume that noise power is zero at all relays while at the receiver we have

E{zz^H} = σ²I= 1

ηI (1)

where z is the circularly-symmetric zero-mean iid Gaussian receiver noise vector. This simplifying non-noisy-relays as- sumption is a first step toward more complete analysis where noisy relays will be considered in future work. (Remark: in [9] a multi-level AF relay network with iid Rayleigh fading is analyzed at high SNR and it is shown that at high SNR the colored noise at destination can be considered as white, which is equivalent to neglecting the noise at relays but not at the destination.) Channel matrices are given by the Kronecker model:

Hi= C^1/2_r,i ΘiC^1/2_t,i i = 1, . . . , N (2) where C_t,i, Cr,iare the transmit and receive correlation matrices and Θ_i are zero-mean iid Gaussian matrices, independent

from each other. Moreover, denoting[Θi]kl the(k, l) entry of Θ_i:

E{|[Θi]kl|²} = 1 ki

i = 1, . . . , N (3) The signal transmitted by the transmitter is x₀ = P0y₀ where y₀ is a circularly-symmetric zero-mean iid Gaussian signal vector such thatE{y0y^H₀ } = I.

Relay at leveli performs linear precoding on its received signal, by multiplying the received vector by a precoding matrix to form its transmitted signal. Therefore the vectors transmitted by the relays x_i, i = 1, . . . , N − 1 are given by:

x_i = Piy_i i = 0, . . . , N − 1 (4) where yi, i = 1, . . . , N − 1 are the signals received at each relaying level, and Pi, are to-be-determined precoding matrices, respecting the per-node average power constraints:

tr(E{xix^H_i }) ≤ Pi i = 0, . . . , N − 1 (5) Amplify-and-Forward is a particular case where the precoding matrix is diagonal.

According to Fig. 1, the signal received at the final receiver can be represented by

y_N = H_NP_{N −1}H_{N −1}P_{N −2}. . . H2P₁H₁P₀y₀+ z

= G_Ny₀+ z (6)

where

G_N =HNP_{N −1}H_{N −1}P_{N −2}. . . H2P₁H₁P₀

=C^1/2_r,NΘNC^1/2_t,NP_{N −1}C^1/2_{r,N −1}Θ_{N −1}C^1/2_{t,N −1}P_{N −2}. . .

× C^1/2_r,2Θ₂C^1/2_t,2P₁C^1/2_r,1Θ₁C^1/2_t,1P₀

(7) We define the matrices :

M_i = C^1/2_t,i+1P_iC^1/2_r,i i = 1, . . . , N − 1 M₀ = C^1/2_t,1P₀

M_N = C^1/2_r,N. (8)

and can then rewrite (7) as:

G_N = MNΘ_NM_{N −1}Θ_{N −1}. . . M2Θ₂M₁Θ₁M₀ (9) The dimensions of the matrices/vectors that are involved in our analysis are given below:

xi: ki× 1 yi: ki× 1 Pi: ki× ki

Hi: ki× k_i−1 Cr,i: ki× ki Ct,i: k_i−1× k_i−1 Θi : ki× k_i−1 Mi: ki× ki

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III. ASYMPTOTICMUTUALINFORMATION

In this section, we derive a closed-form expression for the end-to-end asymptotic instantaneous mutual information between the transmitter and the receiver as the number of antennas in all levels grows large with the same rate. In other words, we find the asymptotic instantaneous mutual information per dimensionI

I = 1 k0

log det(I + ηGNG^H_N) (10) when k0, k1, . . . , kN go to infinity while

ki

kN

→ ρi i = 0, . . . , N (11) The main result is summarized in the following theorem:

Theorem 1: For the system described in section II, as k0, k1, . . . , kN go to infinity while _k^kⁱ

N → ρi, i = 0, . . . , N the asymptotic end-to-end instantaneous mutual information per dimensionI is given by

I = 1 ρ0

N

X

i=0

ρiE

log

1 + η

ρi+1

h^N_i Λi

− Nlog e ρ0

η

N

Y

i=0

hi

(12) whereh0, h1, . . . , hN are the solutions of the system ofN + 1 equations

N

Y

j=0

hj= ρiE

h^N_i Λi

ρi+1+ ηh^N_i Λi

i = 0, . . . , N (13)

and whereE is over Λi with distributionFM^H

i M_i(λ) The proof of this theorem uses tools from free probability theory. After introducing a few transformations and lemmas, we provide hereafter the main steps of the proof of Theorem 1. For the full proof, the reader is referred to [10].

Before giving a sketch of the proof, we would like to point out that this expression of the asymptotic instantaneous mutual information is valid for any arbitrary set of matrices P_i.

Moreover this asymptotic expression depends only on the channel statistics and not on a particular realization of the channel. Thus when the size of the system gets large, by knowing only the statistics of the channel and not the instantaneous channel realization, it is still possible to optimize the instantaneous mutual information, making Theorem 1 a powerful tool for optimizing the system performance. Actually, it is a powerful optimization tool even for a small number of antennas. Indeed as illustrated in section V, experimental results show that the system behaves like in the asymptotic regime even for a small number of antennas. In other words, the asymptotic mutual information can be used to optimize the instantaneous mutual information of a finite-size system when transmitting nodes know only the statistics of the channel (the receiver is assumed to know the channel).

Finally, since the asymptotic mutual information depends only on channel statistics, as long as the statistical properties of the channel matrices do not vary at all hops, the instantaneous mutual information converges asymptotically to the same

deterministic expression for any arbitrary channel realization.

As a consequence, the asymptotic instantaneous end-to-end mutual information is also the asymptotic value of the average end-to-end mutual information.

A. Preliminaries

To prove Theorem 1, we need to introduce the following transformations [6]:

Υ^T(s) ,

Z sλ

1 − sλdF^T(λ) (14)

S^T(z) , z + 1

z Υ⁻¹T (z) , S-transform (15) where T is a Hermitian matrix, F^T is the asymptotic eigenvalue distribution of T as its dimensions go to infinity and Υ⁻¹(Υ(s)) = s.

Finally, we need the following lemmas:

Lemma 1 ( [6, Eq.(15)]): Given an m × n matrix A with

n m= ξ

SAA^H(z) = z + 1 z + ξSA^HA

z ξ

(16) Lemma 2 ( [11]): Let Θ be a zero-mean iid Gaussian matrix and T1 and T2 be Hermitian matrices independent from Θ and with proper dimensions such that ΘT1Θ^HT₂ is meaningful. Then, as the dimensions of the matrices go to infinity, ΘT1Θ^H and T2 become asymptotically free, thus,

SΘT₁Θ^HT₂(z) = SΘT₁Θ^H(z)S^T2(z) (17) Lemma 3 ( [6, Theorem 1]): Consider anm × n matrix B with zero-mean iid entries with variance ¹_n. Assume that the dimensions go to infinity while ^m_n → ζ, then

SBB^H(z) = 1

1 + ζz (18)

B. Main Steps of proof of Theorem 1

The proof of Theorem 1 goes through four steps as follows:

• First step : ObtainSG_NG^H

N(z)

Using Lemmas 1, 3, 2 we show the following theorem:

Theorem 2: As ki, i = 0, . . . , N go to infinity with the same rate, the S-transform of GNG^H_N is given by:

SG_NG^H

N(z) = SM^H NM_N(z)

N

Y

i=1

ρi

z+ ρi−1

SM^H i−1M_i

−1

z ρi−1

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• Second step : UseSGNG^H_N(z) to find ΥGNG^H_N(z) Theorem 3: Let us define ρN +1= 1. We have

sΥ^NG_NG^H

N(s) =

N

Y

i=0

ρi+1Υ⁻¹_M^H

i Mi

ΥG_NG^H

N(s)

ρi

(20)

• Third step : UseΥG_NG^H

N(z) to obtain dI/dη.

First we note that

I = kN

k0

1 kN

kN

X

i=1

log(1 + ηλi(GNG^H_N))

= kN

k0

Z

log(1 + ηλ)dF_G^k^N

NG^H

N

(λ)

a.s.→ 1 ρ0

Z

log(1 + ηλ)dFG_NG^H

N(λ) (21)

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where F_G^k^N_N_G^H

N(λ) is the empirical (non-asymptotic) eigenvalue distribution of GNG^H_N. Due to (21), in the asymptotic regime, the derivative of the mutual information with respect to η is linked to ΥG_NG^H

N(z):

dI

dη = 1

−ρ0η ln 2ΥGNG^H_N(−η). (22) Thus using Theorem 3, we show the following theorem:

Theorem 4: In the asymptotic regime, as k0, k1, . . . , kN

go to infinity while _k^kⁱ

N → ρi, i = 0, . . . , N , the derivative of the instantaneous mutual information is given by:

dI dη = 1

ρ0ln 2

N

Y

i=0

hi (23)

where h0, h1, . . . , hN are the solutions of the system of N + 1 equations

N

Y

j=0

hj= ρiE

h^N_i Λi

ρi+1+ ηh^N_i Λi

i = 0, . . . , N.

(24) The expectation is overΛi with distributionFM^H_i Mi(λ).

• Fourth step : IntegratedI/dη to get I

Since primitive functions of ^dI_dη differ by a constant, the constant was chosen such that the mutual information (12) is null when SNRη goes to zero : lim_η→0I(η) = 0.

IV. OPTIMALTRANSMISSIONSTRATEGY ATSOURCE AND

RELAYS

In previous section, the asymptotic mutual information (12), (13) was derived considering arbitrary precoding matrices Pi, i ∈ {0, . . . , N −1}. In this section, we analyze the optimal linear precoding strategies Pi, i ∈ {0, . . . , N − 1} at source and relays that allow to maximize the mutual information.

We characterize the optimal transmit directions, meaning the singular vectors of the precoding matrices at source and relays, that maximize the asymptotic mutual information. It turns out that those transmit direction are also the ones that maximize the average mutual information of finite size systems, i.e. when k0, k1, . . . , kN are finite. In future work, using the results on the optimal directions of transmission and the asymptotic mutual information (12), (13), we intend to work out the optimal power allocation in the asymptotic regime.

The main result of this section is given by the theorem:

Theorem 5: For i ∈ {1, . . . , N } let Ct,i = Ut,iΛ_t,iU^H_t,i and Cr,i = Ur,iΛr,iU^H_r,i be the eigenvalue decompositions of the covariance matrices Ct,i and Cr,i , where Ut,i and U_r,i are unitary and Λt,i and Λr,i are diagonal, with their respective eigenvalues ordered in decreasing order. Then the optimal linear precoding matrices, that maximize the asymptotic mutual information under power constraints (5) can be written

P₀= Ut,1Λ_P0

P_i= Ut,i+1Λ_PiU^H_r,i , fori ∈ {1, . . . , N − 1} (25)

where ΛPi are diagonal matrices with complex elements.

Moreover, those precoding matrices are also the ones which maximize the average mutual information of a finite size (non- asymptotic) system.

This theorem means that the transmit directions at source and relays maximizing the asymptotic mutual information are such that:

• the source should align the transmit covariance matrix Q= P0P^H₀ on the eigenvectors of the transmit correlation matrix of the channel H₁ to the first relaying layer

• relay i should align the precoding matrix Pi on the eigenvectors U_r,iof receive correlation matrix of channel H_i on the right, and on the eigenvectors U_t,i+1 of the transmit correlation matrix of channel H_i+1 on the left.

• the problem of optimizing P_i can be divided into two decoupled problems: optimizing the transmit directions on one hand, and optimizing the transmit powers on the other hand.

For a detailed proof of Theorem 5, we once again refer the readers to [10]. Nevertheless, we would like to draw the attention of the reader on two points.

First, the proof of this theorem does not rely on the expression of the asymptotic mutual information given in (12) and is independent of Theorem 1. On the contrary, the theorem is first proved for the non-asymptotic regime, and the result is shown to still hold in the limit when the dimensions increase.

Nevertheless by combining Theorem 1 and Theorem 5, the ultimate objective is to find the optimal precoding matrices using only knowledge of the statistics of the channel.

Second, as in the proofs in [12] for the average mutual information of multiple-antenna single-user case with covariance knowledge at transmitter, or in [13] for the average mutual information in the multiple-antenna multi-user case also with covariance knowledge at transmitter, both without relaying, or in [2] for the two-hop relay system with full CSI at the relay, our proof of Theorem 5 is based on Theorem H.1.h in [14].

V. NUMERICALRESULTS

In this section, we present numerical results to validate Theorem 1 and to show that even for a small number of antenna, the behavior of the system is close to the behavior in the asymptotic regime, making Theorem 1 a useful tool for optimization of finite-size systems as well as large networks.

Fig. 2 plots the asymptotic mutual information from Theo- rem 1 as well as the instantaneous mutual information obtained for an arbitrary channel realization (’experimental’ curves) for a system with 10 antennas at transmitter, receiver and each relay level, and 1,2 or 3 hops, the case N = 1 hop corresponding to a MIMO channel. Fig. 3 plots the same type of curves, for a system with 100 antennas at each levels.

Equal power allocation, i.e. matrices Pi proportional to the identity matrix, as well as non-correlated channels, i.e. channel correlation matrices all equal to identity, were considered in these simulations, whose purpose is mainly to validate the formula in Theorem 1, not to optimize the system. We would

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0 5 10 15 20 25 0

0.5 1 1.5 2 2.5 3 3.5 4 4.5

Instantaneous Mutual Information vs SNR − 10 antennas

SNR [dB]

Instantaneous Mutual Information [bits/s]

N = 1, MIMO, Asymptotic N = 1, MIMO, Experimental N = 2, 1 relay, Asymptotic N = 2, 1 relay, Experimental N = 3, 2 relays, Asymptotic N = 3, 2 relays, Experimental

Fig. 2. Asymptotic Mutual Information and Instantaneous Mutual Informa- tion with K = 10 antennas, for MIMO, 1 relay level, and 2 relay levels

like to point out that plotting the experimental curves for different channel realizations gave similar results, and that for the sake of clarity and conciseness, we exhibit the experimental curves only for one realization. Fig. 3 shows the perfect match between the instantaneous mutual information of arbitrary channel realization and the asymptotic mutual information, validating the formula for large dimensions of the network.

On the other hand Fig. 2 shows that the instantaneous mutual information of a system with a small number of antennas behaves very closely to the asymptotic regime, justifying the usefulness of the asymptotic formula even for optimizing systems with small dimensions.

VI. CONCLUSION

We studied a multi-level MIMO relay network, in correlated fading, where relays perform linear precoding on their received signal before retransmission. On one hand, using free probability theory, we derived a closed-form expression of the end-to-end instantaneous mutual information in the asymptotic regime when the number of antennas at all levels goes to infinity with same rate. This expression turns out to depend only on channel statistics and not on particular channel realizations. We also showed that multi-level networks with finite dimensions behave closely to the asymptotic regime, even for a small number of antennas, making the asymptotic mutual information a powerful tool for optimizing the instantaneous mutual information of finite dimensions systems with only statistical knowledge of the channel. On the other hand, we showed that the precoding matrices that maximize the asymptotic mutual information, have a particular form:

the precoding matrices, through their singular vectors, must be aligned on the eigenvectors of the channel transmit and receive correlation matrices. Combining asymptotic mutual information and optimal directions of transmissions, future work will focus on optimizing the power allocations, so as to find the precoding matrices optimizing the mutual information

0 5 10 15 20 25

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Instantaneous Mutual Information vs SNR − 100 antennas

SNR [dB]

Instantaneous Mutual Information [bits/s]

N = 1, MIMO, Asymptotic N = 1, MIMO, Experimental N = 2, 1 relay, Asymptotic N = 2, 1 relay, Experimental N = 3, 2 relays, Asymptotic N = 3, 2 relays, Experimental

Fig. 3. Asymptotic Mutual Information and Instantaneous Mutual Informa- tion with K = 100 antennas, for MIMO, 1 relay level, and 2 relay levels

with only statistical channel knowledge.

ACKNOWLEDGMENT

The authors would like to thank the French Defense Body DGA, BIONETS project (FP6-027748, www.bionets.eu) and Alcatel-Lucent within the Alcatel-Lucent Chair on flexible radio at SUPELEC for supporting this work.

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